A267540 -id:A267540 - cp's OEIS frontend

A267711 Numbers k such that k mod 3 = k mod 5.

0, 1, 2, 15, 16, 17, 30, 31, 32, 45, 46, 47, 60, 61, 62, 75, 76, 77, 90, 91, 92, 105, 106, 107, 120, 121, 122, 135, 136, 137, 150, 151, 152, 165, 166, 167, 180, 181, 182, 195, 196, 197, 210, 211, 212, 225, 226, 227, 240, 241, 242, 255, 256, 257, 270, 271, 272, 285, 286

Offset: 1

Mikk Heidemaa, Jan 19 2016

Periodic differences between the consecutive terms (1,1,13,1,1,13,1,1,13,1,1...).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A267540.

Select[ Range[0, 10000], (Mod[#, 3] == Mod[#, 5]) &]

lista(nn) = for(n=0, nn, if(n%3 == n%5, print1(n, ", "))); \\ Altug Alkan, Jan 19 2016

concat(0, Vec(x^2*(1+x+13*x^2)/((1-x)^2*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jan 28 2016

a(n) = (1/3)*(15*n - 12*cos((2*Pi*n)/3) + 4*sqrt(3)*sin((2*Pi*n)/3) - 27).
G.f.: x^2*(13*x^2+x+1) / ((x-1)^2*(x^2+x+1)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4. - Colin Barker, Jan 28 2016

A267550 Primes p such that p (mod 3) = p (mod 5) = p (mod 7).

Original entry on oeis.org

2, 107, 211, 317, 421, 631, 947, 1051, 1367, 1471, 1787, 1997, 2207, 2311, 2417, 2521, 2731, 2837, 3257, 3361, 3467, 3571, 3677, 4201, 4517, 4621, 4831, 4937, 5147, 5881, 5987, 6091, 6197, 6301, 6827, 7247, 7351, 7457, 7561, 7877, 8087, 8191, 8297, 8821, 9137, 9241, 9661, 9767, 9871, 10501

Offset: 1

Mikk Heidemaa, Jan 17 2016

Or primes p such that p (mod 105) = {1, 2}.

In terms a(4227)...a(4246) their terminal digits alternate: 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A216145, A267540.

[p: p in PrimesUpTo(10000) | p mod 3 eq p mod 5 and p mod 5 eq p mod 7]; // Vincenzo Librandi, Jan 17 2016

Select[ Prime[ Range[10000]], (Mod[#,3] == Mod[#,5] == Mod[#,7]) &](*Or*)
Select[ Prime[ Range[10000]], 0 < Mod[#,105] < 3 &]
Select[Prime[Range[10000]],Length[Union[Mod[#,{3,5,7}]]]==1&] (* Harvey P. Dale, Oct 11 2019 *)

lista(nn) = forprime(p=2, nn, if(p%3 == p%5 && p%5 == p%7, print1(p, ", "))); \\ Altug Alkan, Jan 25 2016

More terms from Vincenzo Librandi, Jan 17 2016

A267781 Expansion of (x(-14x^6 - 32x^5 + 16x^4 + 30x^3 - x + 14) + 17)/((x-1)^2(x+1)).

Original entry on oeis.org

17, 31, 47, 61, 107, 137, 151, 167, 181, 197, 211, 227, 241, 257, 271, 287, 301, 317, 331, 347, 361, 377, 391, 407, 421, 437, 451, 467, 481, 497, 511, 527, 541, 557, 571, 587, 601, 617, 631, 647, 661, 677, 691, 707, 721, 737, 751, 767, 781, 797, 811, 827, 841, 857, 871, 887, 901, 917, 931, 947, 961, 977, 991, 1007

Offset: 1

Mikk Heidemaa, Jan 20 2016

The terms which are primes are the same (p mod 3 = p mod 5) as in A267540 (starting from a(2)=17, their correspondence is verified up to 150000047).

Primes here frequently also have regular intervals and occur mostly in short blocks (consisting of 2-4 primes) rather than singletons, but some blocks can be much longer (e.g., a(1)..a(15) and a(33)..a(43)).

Cf. A063118, A267540.

CoefficientList[ Series[(x*(-14x^6 - 32x^5 + 16x^4 + 30x^3 - x + 14) + 17)/((x - 1)^2*(x + 1)), {x, 0, 63}], x] (* Michael De Vlieger, Jan 21 2016 *)(* Or *)
Flatten @Prepend[ Table[(30*n - (-1)^n + 123)/2, {n, 5, 1000}],{17,31,47,61,107}](* Efficient. Mikk Heidemaa, Jan 21 2016 *)

Vec((x*(-14*x^6-32*x^5+16*x^4+30*x^3-x+14)+17)/((x-1)^2*(x+1)) + O(x^80)) \\ Michel Marcus, Jan 20 2016

a(n) = (30*n - (-1)^n + 123)/2 for n > 4. - Colin Barker, Jan 21 2016

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A267711 Numbers k such that k mod 3 = k mod 5.

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A267550 Primes p such that p (mod 3) = p (mod 5) = p (mod 7).

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A267781 Expansion of (x(-14x^6 - 32x^5 + 16x^4 + 30x^3 - x + 14) + 17)/((x-1)^2(x+1)).

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cp's OEIS Frontend

A267711 Numbers k such that k mod 3 = k mod 5.

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A267550 Primes p such that p (mod 3) = p (mod 5) = p (mod 7).

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Author

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Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A267781 Expansion of (x*(-14x^6 - 32x^5 + 16x^4 + 30x^3 - x + 14) + 17)/((x-1)^2*(x+1)).

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Comments

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PARI

Formula

A267781 Expansion of (x(-14x^6 - 32x^5 + 16x^4 + 30x^3 - x + 14) + 17)/((x-1)^2(x+1)).